Showing papers in "Journal of Applied Probability in 1991"
TL;DR: In this article, the authors introduce a new class of queueing networks in which customers are either "negative" or "positive", and show that this model with exponential service times and Poisson external arrivals, with the usual independence assumptions for service times, and Markovian customer movements between queues, has product form.
Abstract: We introduce a new class of queueing networks in which customers are either 'negative' or 'positive'. A negative customer arriving to a queue reduces the total customer count in that queue by 1 if the queue length is positive; it has no effect at all if the queue length is empty. Negative customers do not receive service. Customers leaving a queue for another one can either become negative or remain positive. Positive customers behave as ordinary queueing network customers and receive service. We show that this model with exponential service times, Poisson external arrivals, with the usual independence assumptions for service times, and Markovian customer movements between queues, has product form. It is quasi-reversible in the usual sense, but not in a broader sense which includes all destructions of customers in the set of departures. The existence and uniqueness of the solutions to the (nonlinear) customer flow equations, and hence of the product form solution, is discussed.
454 citations
TL;DR: In this paper, the authors study single-server queueing models where in addition to regular arriving customers, there are negative arrivals, and the way in which this removal is specified gives rise to several different models.
Abstract: We study single-server queueing models where in addition to regular arriving customers, there are negative arrivals. A negative arrival has the effect of removing a customer from the queue. The way in which this removal is specified gives rise to several different models. Unlike the standard FIFO GI/GI/1 model, the stability conditions for these new models may depend upon more than just the arrival and service rates; the entire distributions of interarrival and service times may be involved.
157 citations
TL;DR: In this article, a new class of life distributions, namely new better than used in convex ordering (NBUC), and its dual, new worse than used for convex order (NWUC), are introduced.
Abstract: A new class of life distributions, namely new better than used in convex ordering (NBUC), and its dual, new worse than used in convex ordering (NWUC), are introduced. Their relations to other classes of life distributions, closure properties under three reliability operations, and heritage properties under shock model and Laplace-Stieltjes transform are discussed.
108 citations
TL;DR: In this paper, the authors present a simpler alternative to the classical proofs for the existence of a stationary (s, S) inventory policy that minimizes the total discounted or average cost over an infinite horizon.
Abstract: The classical proofs for the existence of a stationary (s, S) inventory policy that minimizes the total discounted or average cost over an infinite horizon are lengthy because they depend heavily on the optimality results for corresponding finite-horizon models. This note presents a simpler alternative. Since optimal stationary (s, S) policies are relatively simple to characterize, it is easy to construct a solution to the optimality equation which is satisfied by an (s, S) policy or an equivalent variant thereof. For the discounted model, the proof characterizes an (s, S) policy that is optimal for all initial inventory positions. This policy can be generated by a simple existing algorithm. For the average-cost model, the optimality proof is completed with some additional arguments, which are simple but novel, to overcome the normal difficulties encountered in models with unbounded one-step expected costs.
106 citations
TL;DR: In this article, different choices of distributions for the unobserved covariates are discussed, including binary, gamma, inverse Gaussian and positive stable distributions, which show both qualitative and quantitative differences.
Abstract: Ordinary survival models implicitly assume that all individuals in a group have the same risk of death. It may, however, be relevant to consider the group as heterogeneous, i.e. a mixture of individuals with different risks. For example, after an operation each individual may have constant hazard of death. If risk factors are not included, the group shows decreasing hazard. This offers two fundamentally different interpretations of the same data. For instance, Weibull distributions with shape parameter less than 1 can be generated as mixtures of constant individual hazards. In a proportional hazards model, neglect of a subset of the important covariates leads to biased estimates of the other regression coefficients. Different choices of distributions for the unobserved covariates are discussed, including binary, gamma, inverse Gaussian and positive stable distributions, which show both qualitative and quantitative differences. For instance, the heterogeneity distribution can be either identifiable or unidentifiable. Both mathematical and interpretational consequences of the choice of distribution are considered. Heterogeneity can be evaluated by the variance of the logarithm of the mixture distribution. Examples include occupational mortality, myocardial infarction and diabetes.
84 citations
TL;DR: In this article, the Laplace-smaller-than-C model is used to order two arbitrary life distributions with respect to their Laplace transforms and derive useful inequalities for M/G/1 queues, time series, coherent systems, shock models and cumulative damage models.
Abstract: Two arbitrary life distributions F and G can be ordered with respect to their Laplace transforms. We say F is Laplace-smaller than C if foJ e-sF(t)dt 0. Interpretations of this ordering concept in reliability, operations research, and economics are described. General preservation properties are presented. Using these preservation results we derive useful inequalities and discuss their applications to M/G/1 queues, time series, coherent systems, shock models and cumulative damage models.
83 citations
TL;DR: In this article, the authors examined repairable systems with random lifetime and proposed an age-dependent maintenance strategy which minimizes the total expected discounted cost over an infinite planning horizon, using several properties of the optimal policy derived in this study.
Abstract: In this study we examine repairable systems with random lifetime. Upon failure, a maintenance action, specifying the degree of repair, is taken by a controller. The objective is to determine an age-dependent maintenance strategy which minimizes the total expected discounted cost over an infinite planning horizon. Using several properties of the optimal policy which are derived in this study, we propose analytical and numerical methods for determining the optimal maintenance strategy. In order to obtain a better insight regarding the structure and nature of the optimal policy and to illustrate computational procedures, a numerical example is analysed. The proposed maintenance model outlines a new research channel in the area of reliability with interesting theoretical issues and a wide range of potential applications in various fields such as product design, inventory systems for spare parts, and management of maintenance crews.
69 citations
TL;DR: In this paper, the ergodic properties of birth and death processes are studied and some explicit estimates for the rate of convergence by the methods of theory of differential equations are obtained.
Abstract: The ergodic properties of birth and death processes are studied. We obtain some explicit estimates for the rate of convergence by the methods of theory of differential equations.
69 citations
TL;DR: The notion of strong stochastic convexity (SSCX) was introduced in this article, which is a property enjoyed by a wide range of random variables and is preserved under random mixture, random summation and any increasing and convex operations that are applied to a set of independent random variables.
Abstract: A family of random variables {X(θ)} parameterized by the parameter θ satisfies stochastic convexity (SCX) if and only if for any increasing and convex function f(x), Ef[X(θ)] is convex in θ . This definition, however, has a major drawback for the lack of certain important closure properties. In this paper we establish the notion of strong stochastic convexity (SSCX), which implies SCX. We demonstrate that SSCX is a property enjoyed by a wide range of random variables. We also show that SSCX is preserved under random mixture, random summation, and any increasing and convex operations that are applied to a set of independent random variables. These closure properties greatly facilitate the study of parametric convexity of many stochastic systems. Applications to GI/G/1 queues, tandem and cyclic queues, and tree-like networks are discussed. We also demonstrate the application of SSCX in bounding the performance of certain systems.
61 citations
TL;DR: In this article, it was shown that the stationary, autoregressive, Markovian minification processes introduced by Tavares and Sim can be extended to give processes with marginal distributions other than the exponential and Weibull distributions.
Abstract: It is shown that the stationary, autoregressive, Markovian minification processes introduced by Tavares and Sim can be extended to give processes with marginal distributions other than the exponential and Weibull distributions. Necessary and sufficient conditions on the hazard rate of the marginal distributions are given for a minification process to exist. Results are given for the derivation of the autocorrelation function; these correct the expression for the Weibull given by Sim. Monotonic transformations of the minification processes are also discussed and generate a whole new class of autoregressive processes with fixed marginal distributions. Stationary processes generated by a maximum operation are also introduced and a comparison of three different Markovian processes with uniform marginal distributions is given.
58 citations
TL;DR: In this paper, the authors consider the Markov chain formed by the operation of the move-to-front scheme and show that the eigenvalues of the transition probability matrix are of the form pi, p + y, - - -, Zi1 p, where pi is the probability of selecting the ith item and N is the number of items.
Abstract: In this paper we consider the Markov chain formed by the operation of the move-to-front scheme. We show that the eigenvalues of the transition probability matrix are of the form pi, p + y, - - -, Zi1 p, where pi is the probability of selecting the ith item and N is the number of items; further, that the multiplicity of the eigenvalues of the form Z p, where the summation is over m items is equal to the number of permutations of N - m objects, ordered in some way, such that no object is in its natural position. Finally, we show that the Markov chain is lumpable - many times over. which are to be arranged in some order. At each unit of time, an item is requested - B, being requested, independently of the past, with probability pA, p, > 0, ZN p, = 1. The cost of the retrieval of the requested item is taken to be its ordered position (from the left). A problem of interest is to determine the optimal ordering so as to minimize the long- run average retrieval or search cost. It is obvious that if pi were known, the optimal ordering would be to order the items in decreasing order of the pA's. However, in many situations one does not know the p,'s and one consider schemes which are self-regulating in the sense that if the scheme is operated over a long time the most popular items would tend to be at the left. A number of schemes have been suggested; the two whose properties have been discussed in some details are the move-to-front scheme and the transposition scheme; see Hendricks (1972), (1976), Burville and Kingman (1973), Gonnett et al. (1981). In the move-to-front scheme a requested item when returned is placed at the left end, the other items being moved to the right as necessary to make room for it. In the transposition scheme the requested item exchanges places with the item to its immediate left; if the requested item is at the extreme left, nothing is done. Early work in this area was concerned with the properties of these schemes in the equilibrium state. Here the important result is that of Rivest (1976) who showed that the
TL;DR: In this paper, it was shown that for a sequence of random variables whose failure rates at any time add to 1, the order statistics are stochastically larger than a sample of independent exponential random variables of mean n, but that strong monotone coupling is impossible in general.
Abstract: If Xi, i = 1, ⋯, n are independent exponential random variables with parameters λ1, ⋯, λn, and if Yi, i = 1, ⋯, n are independent exponential random variables with common parameter equal to (λ1 + ⋯ + λn)/n then there is a monotone coupling of the order statistics X(1), ⋯, X(n) and Y(1), ⋯, Y(n); that is, it is possible to construct on a common probability space random variables Xi′, Yi′, i = 1, ⋯, n such that for each i, Y(i)′ ≤ X(i)′ a.s., where the law of the Xi′ (respectively, the Yi′) is the same as the law of the Xi (respectively, the Yi) This result is due to Proschan and Sethuraman, and independently to Ball. We shall here prove an extension to a more general class of distributions for which the failure rate function r(x) is decreasing, and xr(x) is increasing. This very strong order relation allows comparison of properties of epidemic processes where rates of infection are not uniform with the corresponding properties for the homogeneous case. We further prove that for a sequence Zi, i = 1, ⋯, n of independent random variables whose failure rates at any time add to 1, the order statistics are stochastically larger than the order statistics of a sample of n independent exponential random variables of mean n, but that the strong monotone coupling referred to above is impossible in general.
TL;DR: In this article, the authors derived the joint equilibrium distribution of states immediately before and after a batch of customers is released into the network, which can be seen as marginal distributions within a more general framework.
Abstract: Product-form equilibrium distributions in networks of queues in which customers move singly have been known since 1957, when Jackson derived some surprising independence results. A product-form equilibrium distribution has also recently been shown to be valid for certain queueing networks with batch arrivals, batch services and even correlated routing. This paper derives the joint equilibrium distribution of states immediately before and after a batch of customers is released into the network. The results are valid for either discrete- or continuous-time queueing networks: previously obtained results can be seen as marginal distributions within a more general framework. A generalisation of the classical ‘arrival theorem' for continuous-time networks is given, which compares the equilibrium distribution as seen by arrivals to the time-averaged equilibrium distribution.
TL;DR: For the expected discounted reward case, this paper showed that the optimal replacement policy is of the form "replace at the time of the Nth failure" for a stochastically deteriorating system.
Abstract: In this paper, we study a repair replacement model for a stochastically deteriorating system. For the expected discounted reward case, we show that the optimal replacement policy is of the form ‘replace at the time of the Nth failure'.
TL;DR: In this paper, the joint limiting distribution of suitably normalized partial sums and maxima in a stationary strong mixing sequence with finite variance is derived, and it is found that in the limit the two components are independent.
Abstract: The joint limiting distribution of suitably normalized partial sums and maxima in a stationary strong mixing sequence with finite variance is derived. It is found that in the limit the two components are independent. This generalizes Chow and Teugels' result for independent sequences. Motivation for the present study comes from a statistical problem in the analysis of extreme winds.
TL;DR: In this article, the authors studied a single server queue with finite waiting room where the server takes vacations according to two different strategies: (i) an exhaustive service discipline, where server takes a vacation whenever the system becomes empty and these vacations are repeated as long as there are no customers in the system upon return from a vacation.
Abstract: This paper studies a single server queue with finite waiting room where the server takes vacations according to two different strategies: (i) an exhaustive service discipline, where the server takes a vacation whenever the system becomes empty and these vacations are repeated as long as there are no customers in the system upon return from a vacation, i.e. a repeated vacation strategy; (ii) a limited service discipline, where the server begins a vacation either if K customers have been served in the same busy period or if the system is empty and then a repeated vacation strategy is followed. The input process is a general Markovian arrival process introduced by Lucantoni, Meier-Hellstern and Neuts, which as special cases includes the Markov modulated Poisson process and the phase-type renewal process. The service times and vacation times each are generally distributed random variables. For both models, we obtain the queue length distribution at departures, at an arbitrary time instant and at arrival time. We also derive the loss probability of an arriving customer. We obtain formulae for the LST of the virtual waiting time distribution and for the LST of the waiting time distribution at arrival epochs.
TL;DR: In this article, it was shown that the heaps process with Poisson-Dirichlet initial distribution converges to the size-biased permutation of its initial distribution, which leads to a number of new characterizations of the property of invariance under size bias.
Abstract: The heaps process (also known as a Tsetlin library) provides a model for a self-regulating filing system. Items are requested from time to time according to their popularity and returned to the top of the heap after use. The size-biased permutation of a collection of popularities is a particular random permutation of those popularities, which arises naturally in a number of applications and is of independent interest. For a slightly non-standard formulation of the heaps process we prove that it converges to the size-biased permutation of its initial distribution. This leads to a number of new characterizations of the property of invariance under size-biased permutation, notably what might be described as invariance under ‘partial size-biasing' of any order. Finally we consider in detail the heaps process with Poisson–Dirichlet initial distribution, exhibiting the tractable nature of its equilibrium distribution and explicitly calculating a number of quantities of interest.
TL;DR: In this paper, a semi-Pareto process with Pareto inputs is studied and the maximum and minimum of the first n observations are examined as well as the geometric maximum and geometric minimum.
Abstract: Semi-Pareto processes, of which Pareto processes form a proper sub-class, are discussed here. A semi-Pareto process has semi-Pareto inputs. Asymptotic properties of the maximum and minimum of the first n observations are examined as well as the geometric maximum and geometric minimum. A characterization of the semi-Pareto distribution is given. A canonical representation of a special class of Pareto process is also given.
TL;DR: In this article, the stationary distribution of the virtual waiting time, i.i.d. between the arrival and the beginning of service of a customer in a single-server queue, is studied.
Abstract: This paper deals with the stationary distribution of the virtual waiting time, i.e. the time between the arrival and the beginning of service of a customer in a single-server queue that operates as follows. If the server is busy at an arrival time, the customer is rejected. This customer attempts service again after some random delay and continues to do so until the first time at which the server is idle. At this time, the customer is served and leaves the system after service completion. Interarrival times and delays are assumed to be two independent sequences of i.i.d. exponentially distributed random variables. Service times are also i.i.d., generally distributed, and independent of the previous sequences.
TL;DR: The purpose of this paper is to provide some sufficient conditions that imply that as n tends to infinity, the stationary distributions of the truncated chains converge to the stationary distribution of the given chain.
Abstract: We are given a Markov chain with states 0, 1, 2, - .. We want to get a numerical approximation of the steady-state balance equations. To do this, we truncate the chain, keeping the first n states, make the resulting matrix stochastic in some convenient way, and solve the finite system. The purpose of this paper is to provide some sufficient conditions that imply that as n tends to infinity, the stationary distributions of the truncated chains converge to the stationary distribution of the given chain. Our approach is completely probabilistic, and our conditions are given in probabilistic terms. We illustrate how to verify these conditions with five examples.
TL;DR: In this article, the convergence of finite quasi-stationary distributions and a stochastic bound for an infinite quasi-stochastic distribution for absorbing birth-death processes were obtained.
Abstract: Quasi-stationary distributions are considered in their own right, and from the standpoint of finite approximations, for absorbing birth-death processes. Results on convergence of finite quasi-stationary distributions and a stochastic bound for an infinite quasi-stationary distribution are obtained. These results are akin to those of Keilson and Ramaswamy (1984). The methodology is a synthesis of Good (1968) and Cavender (1978).
TL;DR: In this paper, the frequency of a pattern, such as ACGCT, in a long sequence is compared with the expected frequency for all sequences having the same start letter and the same transition counts.
Abstract: Given a realisation of a Markov chain, one can count the numbers of state transitions of each type. One can ask how many realisations are there with these transition counts and the same initial state. Whittle (1955) has answered this question, by finding an explicit though complicated formula, and has also shown that each realisation is equally likely. In the analysis of DNA sequences which comprise letters from the set {A, C, G, T}, it is often useful to count the frequency of a pattern, say ACGCT, in a long sequence and compare this with the expected frequency for all sequences having the same start letter and the same transition counts (or ‘dinucleotide counts' as they are called in the molecular biology literature). To date, no exact method exists; this paper rectifies that deficiency.
TL;DR: In this paper, a multivariate reward process defined on a semi-Markov process is studied, and transform results for the distributions of the reward and related processes are derived through the method of supplementary variables and the Markov renewal equations.
Abstract: A multivariate reward process defined on a semi-Markov process is studied. Transform results for the distributions of the multivariate reward and related processes are derived through the method of supplementary variables and the Markov renewal equations. These transform results enable the asymptotic behavior to be analyzed. A class of first-passage time distributions of the multivariate reward processes is also investigated.
TL;DR: In this paper, the equilibrium solutions of an R-out-of-N system subject to random breakdown are obtained, where there are M spares and a single repairman, who instals good spares into the system when breakdowns occur and also repairs the failed items.
Abstract: We obtain the equilibrium solutions of an R-out-of-N system subject to random breakdown. There are M spares and a single repairman, who instals good spares into the system when breakdowns occur and also repairs the failed items. Installation has pre-emptive repeat priority over repairs. Arbitrary distributions of installation and repair times are allowed. Equilibrium availability and downtime distributions are obtained from the equilibrium state distribution.
TL;DR: In this paper, the Laplace-Stieltjes transforms of the sojourn time distributions of three service disciplines, exhaustive, gated, and 1-limited, are derived for both M/G/1 vacation and polling systems.
Abstract: We study sojourn times in M/G/1 multiple vacation systems and multiqueue cyclic-service (polling) systems with instantaneous Bernoulli feedback. Three service disciplines, exhaustive, gated, and 1-limited, are considered for both M/G/1 vacation and polling systems. The Laplace-Stieltjes transforms of the sojourn time distributions in the three vacation systems are derived. For polling systems, we provide explicit expressions for the mean sojourn times in symmetric cases. Furthermore a pseudo-conservation law with respect to the mean sojourn times is derived for a polling system with a mixture of the three service disciplines.
TL;DR: In this article, it was shown that the expected number of variables to be selected is very close to an easily computable number for general choices of sequences (c n ) and of distributions (F ).
Abstract: Consider an i.i.d. sequence of non-negative random variables ( X 1 , · ··, X n ) with known distribution F. Consider decision rules for selecting a maximum number of the subject to the following constraints: (1) the sum of the elements selected must not exceed a given constant c > 0, and (2) the must be inspected in strict sequence with the decision to accept or reject an element being final at the time it is inspected. Coffman et al. (1987) proved that there exists such a rule that maximizes the expected number E n ( c ) of variables selected, and determined the asymptotics of E n ( c ) for special distributions. Here we determine the asymptotics of E n ( c n ) for very general choices of sequences ( c n ) and of F, by showing that E n ( c ) is very close to an easily computable number. Our proofs are (somewhat deceptively) very simple, and rely on an appropriate stopping-time argument.
TL;DR: For a given transition rate, the uniqueness of the Q-semigroup P(t) = (pu(t)), the recurrence and the positive recurrence of the corresponding Markov chain are three fundamental and classical problems, treated in many textbooks as mentioned in this paper.
Abstract: For a given transition rate, i.e., a Q-matrix Q = (qu) on a countable state space, the uniqueness of the Q-semigroup P(t) = (pu(t)), the recurrence and the positive recurrence of the corresponding Markov chain are three fundamental and classical problems, treated in many textbooks. As an addition, this paper introduces some practical results motivated from the study of a type of interacting particle systems, reaction diffusion processes. The main results are theorems (1. 11), (1.17) and (1.18). Their proofs are quite straightforward.
TL;DR: In this paper, a dynamic notion of mean residual life (MRL) functions in the context of multivariate reliability theory is introduced and its relationship to the multivariate conditional hazard rate functions is studied.
Abstract: In this paper we introduce and study a dynamic notion of mean residual life (mrl) functions in the context of multivariate reliability theory. Basic properties of these functions are derived and their relationship to the multivariate conditional hazard rate functions is studied. A partial ordering, called the mrl ordering, of non-negative random vectors is introduced and its basic properties are presented. Its relationship to stochastic ordering and to other related orderings (such as hazard rate ordering) is pointed out. Using this ordering it is possible to introduce a weak notion of positive dependence of random lifetimes. Some properties of this positive dependence notion are given. Finally, using the mrl ordering, a dynamic notion of multivariate DMRL (decreasing mean residual life) is introduced and studied. The relationship of this multivariate DMRL notion to other notions of dynamic multivariate aging is highlighted in this paper.
TL;DR: In this article, a tractable formula for estimating the intensity of the underlying Poisson process of a Boolean model is given, assuming almost sure boundedness of the primary grain, which is the same assumption as in this paper.
Abstract: A new and practically tractable formula for estimating the intensity of the underlying Poisson process of a Boolean model is given, assuming only almost sure boundedness of the primary grain.
TL;DR: In this article, the limit distribution of the error of approximation of Gaussian stationary periodic processes by random trigonometric polynomials in the uniform metric is investigated, connected with crossings of an increasing level.
Abstract: We consider the limit distribution of maxima and point processes, connected with crossings of an increasing level, for a sequence of Gaussian stationary processes. As an application we investigate the limit distribution of the error of approximation of Gaussian stationary periodic processes by random trigonometric polynomials in the uniform metric. MAXIMUM OF GAUSSIAN PROCESS; POINT PROCESS